I just finished my video as I said, and I followed the tutorial as followed. However, the finished product plays at a horrible frame rate, and the song no longer matches the video. They are not in sync anymore, because of the video's slow playback. Why does the movie playback so slowly??

No, I guess not.
Well, I did take a look at it now, but I'm sooo fuc**** stupid I can't even understand what's being put in front of me. I still don't know what comes next??

All I retained from the tutorial is that I have produced the final copy of my AVI and now it needs compression into a format like Xvid or MPEG-1. Which program is used for this??

Do I "have" to use AVIsynth??

EDIT:
I just tried using the TMPGenc and encoding the movie I had finished. I made an MPEG-1 out of it, but it still isn't crystal clear. If I'm doing something wrong, somebody please help me. The movie is still pretty clear, so I'm done with it.

I will try anything for my next project though, so please...don't hesitate to help me.

mpeg-1 isn't what you want if you're looking for superb visual quality.

if you read the entire guide you'll know what programs to use (but the program you'll probably end up using is virtualdub or virtualdubmod.)
Just follow along and do what the guide says, and if you have a question you should probably search the forums. There's been plenty of threads on the contents of eadfag

if all else fails, play around with stuff and come back to the guide a bit later. It may take a few read-throughs to understand everything.

for the video quality part of it, avisynth really helps. You don't have to use it, but used correctly avisynth can make good video look even better and reduce the filesize of your final encode.

Well I'm not great at avisynth either because it confuses me like hell but compressing it isnt really hard.I normally just use deblock and deen and compress to XVID and it works pretty well.(you can find all this in the guide-Deblock and Deen are cleaning filters)